Lambert-W step-potential
The Lambert-W step-potential affords the fifth – next to those of the harmonic oscillator plus centrifugal, the Coulomb plus inverse square, the Morse, and the inverse square root potentials – exact solution to the stationary one-dimensional Schrödinger equation in terms of the confluent hypergeometric functions. The potential is given as
$$V(x) = \frac{V_0}{1+W (e^{-x/\sigma})}$$. where W is the Lambert function also known as the product logarithm. This is an implicitly elementary function that resolves the equation WeW = z.
The Lambert W-potential is an asymmetric step of height V0 whose steepness and asymmetry are controlled by parameter σ. If the space origin and the energy origin are also included, it presents a four-parametric specification of a more general five-parametric potential which is also solvable in terms of the confluent hypergeometric functions. This generalized potential, however, is a conditionally integrable one (that is, it involves a fixed parameter).
Solution
The general solution of the one-dimensional Schrödinger equation for a particle of mass m and energy E:
$$\frac{d^2\psi}{dx^2}+\frac{2m}{\hbar^2}(E-V(x))\psi=0$$, for the Lambert W-barrier for arbitrary V0 and σ is written as
$$\psi(x)=z^{i\delta/2}e^{-isz/2}\left(\frac{du(z)}{dz}-i\frac{\delta+s}{2}u(z)\right), z=W(e^{-x/\sigma})$$, where u(z) is the general solution of the scaled confluent hypergeometric equation
$$u''(z)+\left(\frac{i\delta}{z}-is\right)u'(z)+\frac{as}{z}u(z)=0$$ and the involved parameters are given as
$$a=\frac{\delta(\delta+s)}{2s}+\frac{\sigma\sqrt{m}V_0}{\sqrt{2E}\hbar}, \delta=2\sigma\sqrt{\frac{2m(E-V_0)}{\hbar^2}}, s=2\sigma\sqrt{\frac{2mE}{\hbar^2}}$$. A peculiarity of the solution is that each of the two fundamental solutions composing the general solution involves a combination of two confluent hypergeometric functions.
If the quantum transmission above the Lambert W-potential is discussed, it is convenient to choose the general solution of the scaled confluent hypergeometric equation as
u = c1(isz)1 − iδ1F1(1+i(a−δ);2−iδ;isz) + c2U(ia;iδ;isz), where c1, 2 are arbitrary constants and 1F1 and U are the Kummer and Tricomi confluent hypergeometric functions, respectively. The two confluent hypergeometric functions are here chosen such that each of them stands for a separate wave moving in a certain direction. For a wave incident from the left, the reflection coefficient written in terms of the standard notations for the wave numbers
$$k_1=\sqrt{\frac{2mE}{\hbar^2}},k_2=\sqrt{\frac{2m(E-V_0)}{\hbar^2}}$$ reads
$$R=e^{-2\pi\sigma k_2}\frac{\sinh{\left(\frac{\pi \sigma}{2 k_1}(k_1-k_2)^2\right)}}{\sinh{\left(\frac{\pi \sigma}{2 k_1}(k_1+k_2)^2\right)}}$$
See also
a/ Confluent hypergeometric potentials
- Quantum harmonic oscillator
- Hydrogen atom
- Morse potential
- Kratzer potential
- Inverse square root potential
b/ Hypergeometric potentials
- Pöschl–Teller potential
- Eckart potential
- Woods-Saxon potential
c/ Other potentials
- Rectangular potential barrier
- Finite potential well
- Infinite potential well
- Delta potential barrier (QM)
- Finite potential barrier (QM)